measures-of-dispersion MCQs for UPSC Prelims

35 practice questions on measures-of-dispersion from the Statistics section of the UPSC Prelims syllabus. 35 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.

12 Easy 17 Medium 6 Hard

Sample questions

Q1
medium

Consider these two statements. I: For any data set, the mean deviation about the median is never greater than the mean deviation about the mean. II: The mean deviation about the median is negative when most of the observations lie below the median. Which is correct?

  1. A Only I is true
  2. B Only II is true
  3. C Both I and II are true
  4. D Neither I nor II is true
Show answer and explanation

Correct answer: A - Only I is true

The sum of absolute deviations is smallest when measured about the median, so the mean deviation about the median cannot exceed the one about the mean, making I true. II is false because absolute deviations are distances and can never be negative, whatever the shape of the data. So the options accepting II fail, and I cannot be rejected.

Q2
easy

Find the mean deviation about the mean of the five observations $4, 7, 8, 9, 12$.

  1. A $2$
  2. B $8$
  3. C $10$
  4. D $0$
Show answer and explanation

Correct answer: A - $2$

The mean is $\frac{40}{5} = 8$. The absolute deviations are $4, 1, 0, 1, 4$, whose sum is $10$, so the mean deviation is $\frac{10}{5} = 2$. The value $0$ comes from adding the deviations with their signs, which always cancel to zero. The value $10$ is the sum of the absolute deviations before dividing, and $8$ is the mean itself.

Q3
medium

A factory tells a buyer that its mean daily output is $500$ units. The buyer also asks for the standard deviation of the daily output. What is the buyer trying to find out?

  1. A How steady the daily output is from day to day
  2. B Whether the reported mean has been computed correctly
  3. C The total output of the factory for the month
  4. D The most frequently occurring daily output
Show answer and explanation

Correct answer: A - How steady the daily output is from day to day

The standard deviation measures how far the daily figures typically lie from their average, so a small value means a dependable, steady supply. The total needs only the mean and the number of days, not the spread. The standard deviation cannot check whether a mean was computed correctly. The most frequent value is the mode, which is a different measure altogether.

Q4
medium

The range of a data set is $28$ and its largest observation is $45$. Find the smallest observation of the data set.

  1. A $73$
  2. B $-17$
  3. C $17$
  4. D $28$
Show answer and explanation

Correct answer: C - $17$

Range $=$ largest $-$ smallest, so smallest $= 45 - 28 = 17$. The value $-17$ comes from subtracting the largest observation from the range instead of the other way round. The value $73$ adds the range to the largest observation instead of subtracting it. The value $28$ merely repeats the range, which is a spread and not an observation.

Q5
easy

Assertion (A): The range of the data $7, 7, 7, 7, 7$ is $0$. Reason (R): The range of a data set is zero exactly when all its observations are equal.

  1. A Both A and R are true, and R is the correct explanation of A
  2. B Both A and R are true, but R is not the correct explanation of A
  3. C A is true, but R is false
  4. D A is false, but R is true
Show answer and explanation

Correct answer: A - Both A and R are true, and R is the correct explanation of A

Here the largest and the smallest observation are both $7$, so the range is $7 - 7 = 0$ and A is true. R is also true: the range is zero only if the maximum equals the minimum, which forces every observation to be the same value, and R is exactly why A holds. So the options that call A false or R false are both wrong, and the second option fails because R does explain A.

Q6
easy

Assertion (A): Two data sets that have the same mean must also have the same standard deviation. Reason (R): The mean describes where the data is centred, while the standard deviation describes how the observations scatter about that centre.

  1. A Both A and R are true, and R is the correct explanation of A
  2. B Both A and R are true, but R is not the correct explanation of A
  3. C A is true, but R is false
  4. D A is false, but R is true
Show answer and explanation

Correct answer: D - A is false, but R is true

A is false: $4, 5, 6$ and $1, 5, 9$ both have mean $5$ but very different spreads. R is a correct description of what the two measures do, and it is precisely why equal means say nothing about the standard deviation. So the two options that call A true fail, and R cannot be marked false.

Q7
hard

The range of a set of eight observations is $24$. Every observation is multiplied by $3$ and then $5$ is subtracted from each result. Find the range of the new set of observations.

  1. A $72$
  2. B $77$
  3. C $24$
  4. D $67$
Show answer and explanation

Correct answer: A - $72$

If $x$ becomes $3x - 5$, then the largest and smallest values are transformed the same way, and their difference becomes $3$ times the old difference: $3 \times 24 = 72$. The subtraction of $5$ shifts both extremes equally and cancels out, so $67$ and $77$ wrongly apply the shift to the range. The value $24$ ignores the multiplication, which does change the spread.

Q8
medium

To find the mean deviation about the median of a set of raw observations, what must be done before any deviation is written down?

  1. A Subtract the smallest observation from each of the others
  2. B Find the arithmetic mean of the observations
  3. C Arrange the observations in ascending order and locate the middle value
  4. D Count how many distinct values the data set contains
Show answer and explanation

Correct answer: C - Arrange the observations in ascending order and locate the middle value

The median is defined only for ordered data, so the observations are sorted first and the middle value (or the average of the two middle values) is taken. The mean is not needed at all when deviations are measured about the median. Subtracting the smallest value gives deviations about the minimum, not the median. Counting distinct values plays no part in the calculation.

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